425 research outputs found
Compressed Sensing and Affine Rank Minimization under Restricted Isometry
This paper establishes new restricted isometry conditions for compressed
sensing and affine rank minimization. It is shown for compressed sensing that
guarantees the exact recovery of all
sparse signals in the noiseless case through the constrained
minimization. Furthermore, the upper bound 1 is sharp in the sense that for any
, the condition is
not sufficient to guarantee such exact recovery using any recovery method.
Similarly, for affine rank minimization, if
then all matrices with
rank at most can be reconstructed exactly in the noiseless case via the
constrained nuclear norm minimization; and for any ,
does not ensure
such exact recovery using any method. Moreover, in the noisy case the
conditions and
are also sufficient for
the stable recovery of sparse signals and low-rank matrices respectively.
Applications and extensions are also discussed.Comment: to appear in IEEE Transactions on Signal Processin
Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices
This paper considers compressed sensing and affine rank minimization in both
noiseless and noisy cases and establishes sharp restricted isometry conditions
for sparse signal and low-rank matrix recovery. The analysis relies on a key
technical tool which represents points in a polytope by convex combinations of
sparse vectors. The technique is elementary while leads to sharp results.
It is shown that for any given constant , in compressed sensing
guarantees the exact recovery of all
sparse signals in the noiseless case through the constrained
minimization, and similarly in affine rank minimization
ensures the exact reconstruction of
all matrices with rank at most in the noiseless case via the constrained
nuclear norm minimization. Moreover, for any ,
is not sufficient to guarantee
the exact recovery of all -sparse signals for large . Similar result also
holds for matrix recovery. In addition, the conditions and are also shown to
be sufficient respectively for stable recovery of approximately sparse signals
and low-rank matrices in the noisy case.Comment: to appear in IEEE Transactions on Information Theor
Guaranteed Rank Minimization via Singular Value Projection
Minimizing the rank of a matrix subject to affine constraints is a
fundamental problem with many important applications in machine learning and
statistics. In this paper we propose a simple and fast algorithm SVP (Singular
Value Projection) for rank minimization with affine constraints (ARMP) and show
that SVP recovers the minimum rank solution for affine constraints that satisfy
the "restricted isometry property" and show robustness of our method to noise.
Our results improve upon a recent breakthrough by Recht, Fazel and Parillo
(RFP07) and Lee and Bresler (LB09) in three significant ways:
1) our method (SVP) is significantly simpler to analyze and easier to
implement,
2) we give recovery guarantees under strictly weaker isometry assumptions
3) we give geometric convergence guarantees for SVP even in presense of noise
and, as demonstrated empirically, SVP is significantly faster on real-world and
synthetic problems.
In addition, we address the practically important problem of low-rank matrix
completion (MCP), which can be seen as a special case of ARMP. We empirically
demonstrate that our algorithm recovers low-rank incoherent matrices from an
almost optimal number of uniformly sampled entries. We make partial progress
towards proving exact recovery and provide some intuition for the strong
performance of SVP applied to matrix completion by showing a more restricted
isometry property. Our algorithm outperforms existing methods, such as those of
\cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion
problem by an order of magnitude and is also significantly more robust to
noise.Comment: An earlier version of this paper was submitted to NIPS-2009 on June
5, 200
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
The affine rank minimization problem consists of finding a matrix of minimum
rank that satisfies a given system of linear equality constraints. Such
problems have appeared in the literature of a diverse set of fields including
system identification and control, Euclidean embedding, and collaborative
filtering. Although specific instances can often be solved with specialized
algorithms, the general affine rank minimization problem is NP-hard. In this
paper, we show that if a certain restricted isometry property holds for the
linear transformation defining the constraints, the minimum rank solution can
be recovered by solving a convex optimization problem, namely the minimization
of the nuclear norm over the given affine space. We present several random
ensembles of equations where the restricted isometry property holds with
overwhelming probability. The techniques used in our analysis have strong
parallels in the compressed sensing framework. We discuss how affine rank
minimization generalizes this pre-existing concept and outline a dictionary
relating concepts from cardinality minimization to those of rank minimization
Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery
This paper establishes a sharp condition on the restricted isometry property
(RIP) for both the sparse signal recovery and low-rank matrix recovery. It is
shown that if the measurement matrix satisfies the RIP condition
, then all -sparse signals can be recovered exactly
via the constrained minimization based on . Similarly, if
the linear map satisfies the RIP condition ,
then all matrices of rank at most can be recovered exactly via the
constrained nuclear norm minimization based on . Furthermore, in
both cases it is not possible to do so in general when the condition does not
hold. In addition, noisy cases are considered and oracle inequalities are given
under the sharp RIP condition.Comment: to appear in Applied and Computational Harmonic Analysis (2012
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